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Abstract

We present an efficient and faithful hyperentanglement purification protocol (hyper-EPP) for three-photon system in mixed hyperentangled Greenberger-Horne-Zeilinger states with bit-flip errors in both spatial-mode and polarization degrees of freedom (DOFs), resorting to the fidelity-robust quantum gates and hyperentanglement link. Our high-efficiency hyper-EPP comes from two aspects. One is to pump the higher-fidelity hyperentanglement from different three-photon systems into the same three-photon system with fidelity-robust swap gates, the other is to reproduce some hyperentangled three-photon systems from hyperentangled two-photon subsystems based on hyperentanglement link. Moreover, as the infidelity originating from imperfect single-photon scattering can be heralded as a failure by triggering a detector, our hyper-EPP operates faithfully with the present quantum circuits. Furthermore, our hyper-EPP can be directly extended to purify multiple photon systems entangled in one DOF or hyperentangled in multiple DOFs.

Figures (7)

Fig. 1. (a) Schematic diagrams of an NV-center-cavity system, and the optical transitions of the NV center with the circularly polarized photons. $R^{\uparrow } (R^{\downarrow })$ and $L^{\uparrow } (L^{\downarrow })$ represent the right- and left-circularly polarized photons propagating along (against) the quantization axis $z$, respectively. (b) Schematic diagram of a modified NV union. SW is an optical switch, which makes the photons entering into and going out of the circuit unit in sequence. M is a mirror and D is a single-photon detector. BS is a $50:50$ beam splitter, which performs the Hadamard operation, that is, $|i_{1}\rangle \rightarrow (|j_{1}\rangle +|j_{2}\rangle )/\sqrt {2}$, or $|i_{2}\rangle \rightarrow (|j_{1}\rangle -|j_{2}\rangle )/\sqrt {2}$, in the spatial-mode DOF of one photon. H represents a quarter-wave plate, which performs the Hadamard operation, that is, $|R\rangle \rightarrow (|R\rangle +|L\rangle )/\sqrt {2}$, or $|L\rangle \rightarrow (|R\rangle -|L\rangle )/\sqrt {2}$, in the polarization DOF of one photon.

Fig. 2. Schematic diagram of the fidelity-robust spatial-polarization parity-check gate (S-P-PCG) for a two-photon system. CPBS is a circularly polarizing beam splitter, which reflects the left-circular-polarization photon $\vert L\rangle$ and transmits the right-circular-polarization photon $\vert R\rangle$, respectively. X is a half-wave plate, which performs a bit-flip operation on the polarization DOF of the photon. T is a partially transmitting mirror.

Fig. 3. (a) Schematic diagram of the fidelity-robust spatial-spatial-swap (S-S swap ) gate. (b) Schematic diagram of the fidelity-robust polarization-polarization-swap (P-P-swap) gate. The red quantum circuit represent that the photons will enter again for the second round.